Interpolatory projection methods for structure-preserving model reduction

Beattie, Christopher; Gugercin, Serkan

doi:10.1016/j.sysconle.2008.10.016

Cited by 137 publications

(181 citation statements)

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“…In addition to these examples of parametric model reduction, several studies have compared various surrogate modeling approaches in the context of specific examples. For example, [32] shows that in a two-dimensional driven cavity flow example for a viscoelastic material, a projection-based reduced model with 60 degrees of freedom performs significantly better than a coarser discretization with 14,803 degrees of freedom. The paper [100] compares projection-based reduced models to stochastic spectral approximations in a statistical inverse problem setting and concludes that, for an elliptic problem with low parameter dimension, the reduced model requires fewer offline simulations to achieve a desired level of accuracy, while the polynomial chaos-based surrogate is cheaper to evaluate in the online phase.…”

Section: Parametric Model Reduction In Actionmentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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Abstract. Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and large-scale systems of parameterized ordinary differential equations. The goal of parametric model reduction is to generate low-cost but accurate models that characterize system response for different values of the parameters. This paper surveys state-of-the-art methods in projection-based parametric model reduction, describing the different approaches within each class of methods for handling parametric variation and providing a comparative discussion that lends insights to potential advantages and disadvantages in applying each of the methods. We highlight the important role played by parametric model reduction in design, control, optimization, and uncertainty quantification-settings that require repeated model evaluations over different parameter values.

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Section: Parametric Model Reduction In Actionmentioning

confidence: 99%

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

2015

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“…Although our focus here is on data-driven interpolation, we revisit briefly the structurepreserving interpolatory projection framework introduced in [7] and establish a connection with realizations arising from Corollary 3.6. Theorem 3.21 (Structure-preserving interpolatory projection [7]).…”

Section: Connection To Structure-preserving Interpolatory Projectionsmentioning

confidence: 99%

“…Theorem 3.21 (Structure-preserving interpolatory projection [7]). Consider the generalized realization H(s) = C(s)K(s) −1 B(s) where both C(s) ∈ C p×N and B(s) ∈ C N ×m are analytic in the right half plane and K(s) ∈ C N ×N is analytic and full rank throughout the right half plane.…”

Section: Connection To Structure-preserving Interpolatory Projectionsmentioning

confidence: 99%

Data-driven structured realization

Schulze

Unger

Beattie

et al. 2018

Linear Algebra and its Applications

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We present a framework for constructing structured realizations of linear dynamical systems having transfer functions of the form C(−1 B where h 1 , h 2 , ..., h K are prescribed functions that specify the surmised structure of the model. Our construction is data-driven in the sense that an interpolant is derived entirely from measurements of a transfer function. Our approach extends the Loewner realization framework to more general system structure that includes second-order (and higher) systems as well as systems with internal delays. Numerical examples demonstrate the advantages of this approach.

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“…For the solution of this problem in the projection framework for G(s) = C(sE − A) −1 B, see Gallivan et al [2005]. Beattie and Gugercin [2009] recently extended the tangential interpolation problem to a much more general setting where G(s) has a generalized coprime factorization, G(s) = C(s)K(s) −1 B(s) with B(s), C(s), and K(s) given as meromorphic matrix-valued functions. Indeed, we use this setting to solve the weighted interpolatory model reduction problem.…”

Section: Interpolatory Projections For Weighted Model Reductionmentioning

confidence: 99%

“…Proof: The proof uses the generalized interpolation setting of Beattie and Gugercin [2009] for the transfer functions…”

Section: Interpolatory Projections For Weighted Model Reductionmentioning

confidence: 99%